Necessary conditions for the compensation approach for a random walk in the quarter-plane

Yanting Chen*, Richard J. Boucherie, Jasper Goseling

*Corresponding author for this work

Research output: Contribution to journalArticleAcademicpeer-review

Abstract

We consider the invariant measure of homogeneous random walks in the quarter-plane. In particular, we consider measures that can be expressed as a countably infinite sum of geometric terms which individually satisfy the interior balance equations. We demonstrate that the compensation approach is the only method that may lead to such a type of invariant measure. In particular, we show that if a countably infinite sum of geometric terms is an invariant measure, then the geometric terms in an invariant measure must be the union of at most six pairwise-coupled sets of countably infinite cardinality each. We further show that for such invariant measure to be a countably infinite sum of geometric terms, the random walk cannot have transitions to the north, northeast or east. Finally, we show that for a countably infinite weighted sum of geometric terms to be an invariant measure at least one of the weights must be negative.

Original languageEnglish
Number of pages20
JournalQueueing systems
DOIs
Publication statusE-pub ahead of print/First online - 8 Jul 2019

Fingerprint

Invariant Measure
Infinite sum
Random walk
Necessary Conditions
Term
Balance Equations
Weighted Sums
Compensation and Redress
Pairwise
Cardinality
Interior
Union
Demonstrate

Keywords

  • UT-Hybrid-D
  • Compensation approach
  • Geometric term
  • Invariant measure
  • Pairwise-coupled
  • Quarter-plane
  • Random walk
  • Algebraic curve

Cite this

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title = "Necessary conditions for the compensation approach for a random walk in the quarter-plane",
abstract = "We consider the invariant measure of homogeneous random walks in the quarter-plane. In particular, we consider measures that can be expressed as a countably infinite sum of geometric terms which individually satisfy the interior balance equations. We demonstrate that the compensation approach is the only method that may lead to such a type of invariant measure. In particular, we show that if a countably infinite sum of geometric terms is an invariant measure, then the geometric terms in an invariant measure must be the union of at most six pairwise-coupled sets of countably infinite cardinality each. We further show that for such invariant measure to be a countably infinite sum of geometric terms, the random walk cannot have transitions to the north, northeast or east. Finally, we show that for a countably infinite weighted sum of geometric terms to be an invariant measure at least one of the weights must be negative.",
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Necessary conditions for the compensation approach for a random walk in the quarter-plane. / Chen, Yanting; Boucherie, Richard J.; Goseling, Jasper.

In: Queueing systems, 08.07.2019.

Research output: Contribution to journalArticleAcademicpeer-review

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N2 - We consider the invariant measure of homogeneous random walks in the quarter-plane. In particular, we consider measures that can be expressed as a countably infinite sum of geometric terms which individually satisfy the interior balance equations. We demonstrate that the compensation approach is the only method that may lead to such a type of invariant measure. In particular, we show that if a countably infinite sum of geometric terms is an invariant measure, then the geometric terms in an invariant measure must be the union of at most six pairwise-coupled sets of countably infinite cardinality each. We further show that for such invariant measure to be a countably infinite sum of geometric terms, the random walk cannot have transitions to the north, northeast or east. Finally, we show that for a countably infinite weighted sum of geometric terms to be an invariant measure at least one of the weights must be negative.

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